By the Birkhoff ergodic theorem,
the irregular set is
universally null:
\(\mu (I_\varphi (X,T))=\mu (I(X, T))=0\)  for every
\(T\) -invariant Borel probability measure \(\mu \)  on \(X\)  and \(\varphi \in C_b(X)\) . The
complement of the irregular set, denoted \(Q(X,T)\) , is called the
quasi-regular set of \((X,T)\) .
If \(X\)  is compact, then it is easy to see that every quasi-regular point must be also generic for some \(T\) -invariant measure (cf. [1]}).
It seems to be a little known fact that each quasi-regular point must be generic for some \(T\) -invariant measure also in the general situation, that is, for every Polish metric space.
For example, Oxtoby in [1]} (following Fomin) adds an additional condition to the definition of quasi-regular point for a Borel automorphism to get a similar result.
